Negative cross effect

As was mentioned earlier, deterministic models of chemical reactions might be identified with eqn (1.3). However, not all kinds of systems of differential equations, not even all those with a polynomial right-hand side can be considered as reaction kinetics equations. Trivially, the term -kc2 t)c t) cannot occur in a rate equation referring to the velocity of Cj, since the quantity of a component cannot be reduced in a reaction in which the component in question does not take place. Putting it another way, the negative cross-effect is excluded. A necessary and sufficient condition is required to restrict eqn (1.3) to be able to be a kinetic equation. 

Another kind of approach has been initiated in the recent investigations by Toth and Hars (1986a). They studied linear transforms of the Lorenz equation (1968) and of the Rossler model (1976) in order to obtain kinetic models. The failure of their efforts underline the importance of negative cross-effects. Based upon these results the following conjecture can be formulated if a nonkinetic polynomial differential equation shows chaotic behaviour then it cannot be transformed into a kinetic one. 

It is an astonishing fact that the converse of the theorem holds as well. If the right-hand side of a differential equation is an (M, M)-polynomial without negative cross-effects then it may be considered as the induced kinetic differential equation of a reaction, or, in other words, if there is no negative cross-effect in the right-hand side then there exists a reaction with the given equation as its deterministic model. 

Polynomial differential equations without negative cross-effects will usually be called kinetic differential equations from now on. 

Is the lack of negative cross-effects not too strong a restriction in the sense that a randomly selected polynomial differential equation is usually nonkinetic. How dense is the set of kinetic differential equations within the set of polynomial ones ... 

The property that the vector on the right-hand side of a differential equation points into the interior of the first orthant is obviously a necessary condition for negative cross-effects to be absent. Give a polynomial example showing that it is not sufficient. 

It is also shown that a kinetic equation remains a kinetic one if transformed by a positive definite diagonal transformation /change of scale/ and any natural transformation of time does not change the presence or absence of negative cross-effects. 

Based upon this results we strongly believe that impossibility to eliminate negative cross-effect from equations showing chaotic behaviour may prove an important, if not characteristic, property. Nevertheless, it should be mentioned that kinetic equations with apparent /numerically demonstrated/ chaotic behaviour do exist Dl, although most examples contain negative cross-effects Cd. 

Generally these globular dendritic architectures offer several advantages over other kinds of organic polymers, such as the full exposure of the catalytic centers to the environment. In contrast to linear or cross-Hnked polymeric supports, which can partially hide catalytic centers, the functional groups are located on the surface of the dendritic nanoparticle and diffusional Hmitations are less relevant Furthermore the close proximity of the catalytic centers on the surface of the dendritic polymer can enhance the catalytic activity by multiple complexation or even cooperativity. This behavior is described as positive dendritic effect. However, in some cases a negative dendritic effect was observed, which is caused by an undesired interaction or electron transfer between the neighboring catalytic centers on the surface of the dendrimer [70]. 

Eq. 2.18) and Ohm s law for electrical conduction (where the electrical conductivity is also always positive, according to Exercise 2.1 s solution). However, this is not necessarily the case for the cross-effect tensors /3 and 7. For example, in mass diffusion in a thermal gradient, the heat of transport can be either positive or negative the direction of the atom flux in a temperature gradient can then be in either direction. The anisotropic equivalent to the heat of transport relates the direction of the mass diffusion to the direction of the temperature gradient. There is no physical requirement that these quantities could not be in reversed directions, and indeed, sometimes they are. 

Effective diffiisivities D,j control the diffusion flux of component i, coupled with its concentration gradient. The effective diffusion crossefficients D,j (i j) characterize the cross effect (or the interdiffiision interference of the two components) thereby defining the diffusion of given component i associated with the concentration gradient of the other component j. The cross diffiisivities are small compared to the diagonal ones and may, in principle, be negative and tend to zero while tend to zero. 

If the tertiary pr-MDI and bu-MDI gain access to the interior of the cell to exert their calcium antagonistic actions, it would be predicted that their quaternary ammonium analogues would be inactive due to their exclusion from the intracellular compartment as a result of their inability to cross biological membranes. As predicted, the tertiary, but not the quaternary, MDIs produce a negative inotropic effect on the isolated electrically-driven guinea pig left atrium (42) and inhibit potassium-induced and norepinephrine-induced contractions of the isolated rat aortic strip (43). 

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